Can NEAT produce neural networks where inputs are directly connected to outputs?
Yes, it is possible (depending on the nature of your problem), using the four types of standard NEAT mutation, but it is improbable.

When the NEAT algorithm begins, it operates on a blank canvas. After each generation, the algorithm will either:


Construct a new axon
Construct a new node on an existing axon
Update existing weights/bias
Remove a node or axon from the network


(However, in general, NEAT does not produce neural networks where two input (or output) nodes are connected.)